Boundary Layer Criterion for the Low Mach Number and Inviscid Limits in Compressible Fluid-Rigid Body Interaction

Abstract

We investigate the simultaneous low Mach number and vanishing viscosity limit for a three-dimensional compressible viscous fluid interacting with a freely moving rigid body. The fluid satisfies the compressible Navier-Stokes equations and the no-slip condition on the body surface, while the body evolves according to Newton's laws driven by the fluid stresses. We establish convergence of finite-energy weak solutions to a strong solution of the corresponding incompressible Euler-rigid body system on its interval of existence. Assuming that the initial relative energy tends to zero, convergence holds under a Kato-type condition in a boundary layer whose thickness is proportional to the viscosity. This condition combines vanishing viscous dissipation with a weighted estimate on the fluid momentum relative to the rigid body motion. The proof uses the relative entropy method and a boundary layer corrector adapted to the fluid-rigid body coupling, providing a sufficient condition for convergence in relative energy as the Mach number and viscosity tend to zero simultaneously.